
which is obtained by putting \(Pr=0\)  in (REF ).
The larger kernel (REF ) indicates that the degeneracy is stronger in the vanishing Prantl number regime since the dissipativity of \(L_{Pr}\)  stops operating on the null space.
In the case \(Pr>0\) , the full dissipativity of \(L_{Pr}\)  can be recovered by the standard argument [1]}, [2]}, [3]}: the derivatives of the kernels are estimated using the micro-macro equations that govern the evolution of the degenerate part, and the lowest order estimates are derived by combining the derivative estimates and the Poincaré inequality with the vanishing moments of the perturbation up to the second order.
On the other hand, when the Prandtl number vanishes, a novel difficulty unobserved in the previous literature arises: The micro-macro system now involves a non-conservative quantity, namely the heat flux part. This leads to a more complicated micro-macro system, and more seriously, the lowest order estimate of these new terms can not be treated in a similar manner as in the previous case, since the third-order moment of the perturbation related to the heat flux does not vanish. To overcome this, we assume that the total third moment of the initial data vanishes:
\(\int _{\mathbb {T}^3\times \mathbb {R}^3} F_0(x,v)v_i|v|^2dvdx= 0.\) 
